# Questions on rationality and quartic threefolds-Alena Pirutka

**Abstract:** In a context of the Lüroth problem, for `K` a function field of a smooth projective variety `X` over a field `k`, one can ask whether `K` is a purely transcendental over `k` (resp. a subfield of a purely transcendental extension of `k`, resp. becomes purely transcendental after adding some independent variables), that is, if `X` is rational (resp. unirational, resp. stably rational). In this talk we will discuss various invariants which allows to answer this questions for some classes of varieties, and more specifically, for quartic threefolds. By a celebrated result of Iskovskikh and Manin, no smooth quartic hypersurface in **P**^{4}_{C} is rational. Using a specialisation method introduced by C. Voisin, as well as a method based on the universal properties of the Chow group of zero-cycles, we will show that a lot of such quartics are not stably rational. This is a joint work with J.-L. Colliot-Thélène.